Question

Proof that the violation of the Kelvin-Planck statement leads to the violation of the Clausius statement.

Answer 1

assume that Kelvin-Planck statement is incorrect.

Consider a cyclically working device 1, which absorbs energy Q1 as heat from a thermal reservoir at TH. Equivalent amount of work W(W=Q1) is performed.

Consider another device 2 operating as a cycle, which absorbs energy QL as heat from a low temperature thermal reservoir at TL and rejects energy QH (QH=QL+W). Such a device does not violate Clausius statement.

If the two devices are now combined, the combined device (enclosed by the dotted boundary) transfers heat QL from the low temperature reservoir at TL to a high temperature reservoir at TH with out receiving any aid from an external agent, which is the violation of the Clausius statement.

Likewise let us assume that the Clausius statement is incorrect. So we have a device 1, cyclically working transferring heat Q from a low temperature reservoir at TL to a high temperature thermal reservoir at TH . Consider another device 2, which absorbs heat Q1 from a high temperature reservoir at TH does work W and rejects energy Q as heat tot the low temperature reservoir at TL as shown in figure.

If the two devices are combined (shown in figure by a dotted enclosure), then the combined device receives energy (Q1-Q) as heat from a thermal reservoir and delivers equivalent work (W=Q1-Q) in violation of the Kelvin-Planck statement.

Therefore violation of Clausius statement leads to the violation of the Kelvin-Planck statement. Hence, these two statements are equivalent.